El e c t ro n ic
Jo ur
n a l o
f P
r o
b a b il i t y
Vol. 14 (2009), Paper no. 63, pages 1863–1883. Journal URL
http://www.math.washington.edu/~ejpecp/
Nonlinear filtering with signal dependent observation
noise
Dan Crisan
∗, Michael Kouritzin
†and Jie Xiong
‡Abstract
The paper studies the filtering problem for a non-classical framework: we assume that the ob-servation equation is driven by a signal dependent noise. We show that the support of the conditional distribution of the signal is on the corresponding level set of the derivative of the quadratic variation process. Depending on the intrinsic dimension of the noise, we distinguish two cases: In the first case, the conditional distribution has discrete support and we deduce an explicit representation for the conditional distribution. In the second case, the filtering problem is equivalent to a classical one defined on a manifold and we deduce the evolution equation of the conditional distribution. The results are applied to the filtering problem where the observa-tion noise is an Ornstein-Uhlenbeck process.
Key words: Nonlinear Filtering, Ornstein Uhlenbeck Noise, Signal-Dependent Noise. AMS 2000 Subject Classification:Primary 60G35; Secondary: 60G30.
Submitted to EJP on October 9, 2007, final version accepted August 10, 2009.
∗Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2BZ, UK
(d.crisan@imperial.ac.uk)
†Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada (mkouritz@math.ualberta.ca). Research supported by NSERC.
1
Introduction
Let(Ω,F,P)be a probability space on which we have defined a homogeneous Markov process X. We can obtain information onX by observing an associated process y which is a function ofX plus random noisen:
yt=h(Xt) +nt. (1.1)
In most of the literaturent is modeled by white noise, which does not exist in the ordinary sense, but rather as a distribution of a generalized derivative of a Brownian motion. That is, the process W defined formally as
Wt=
Z t
0
nsds, t≥0,
is a Brownian motion and the (integrated) observation model turns into one of the (classical) form
Yt =
Z t
0 ysds
=
Z t
0
h(Xs)ds+Wt,
whereY is the accumulated observation. In this case, the observationσ-field isFtY =σAug{Ys, s≤
t}and the desired conditional distribution
πt(·)=. P(Xt∈ ·|FtY), t≥0,
satisfies the Kushner-Stratonovich or Fujisaki-Kallianpur-Kunita equation
dπt(f) =πt(L f)d t+ (πt(f hT)−πt(f)πt(hT))dνˆt, (1.2)
where Lis the generator ofX andνˆ, defined as
ˆ
νt=Yt−
Z t
0
πs(h)ds, t≥0,
is a Brownian motion called the innovation process.
Balakrishnan (see Kallianpur and Karandikar[8], p. 3) states that this (integrated) approach is not suitable for application since the results obtained cannot be instrumented. Kunita[9], Mandal and Mandrekar [11] and Gawarecki and Mandrekar [4]studied the model (1.1) when nis a general Gaussian process. The most important example is the case whennis an Ornstein-Uhlenbeck process given by
dOt=−βOtd t+βdWt, (1.3)
andW is a standardm-dimensional Wiener process. The observation model becomes
yt=h(Xt) +Ot, t≥0 (1.4)
or the integral form
Yt=
Z t
0
h(Xs)ds+
Z t
0
To fix the ideas, we let the signalX be aRd-valued process given by
d Xt =b(Xt)d t+c(Xt)d Bt, t≥0
where b, c areRd, respectively Rd×d-valued functions and B is a standard d-dimensional Wiener process independent ofW. The optimal filterπis given by
πtf =E(f(Xt)|FtY).
In the aforementioned papers, a Kallianpur-Striebel formula is given, the filtering equation forπis derived and it is proved thatπconverges to the solution of the (classical) FKK equation asβ tends to infinity. However, the conditions imposed in these papers[9],[11],[4]are very restrictive. Most notably, the authors assume that the map t 7→ h(Xt) is differentiable. To remove this restrictive condition, Bhatt and Karandikar[1]consider the variant observation model
yt=α
Z t
(t−α−1)∨0
h(Xs)ds+Ot
forα >0 and obtain the same results for this modified model.
In this paper, we deal with the original model (1.4) but no longer assume differentiability of the map t7→h(Xt). As we will see, this will lead to an observation model with signal dependent observation noise
d Yt=h(Xt)d t+σ(Xt)dWt.
In turn, the filtering problem with signal dependent observation noise will be converted into a clas-sical one (with signal independent observation noise) via a suitably chosen stochastic flow mapping. This article is organized as follows: In Section 2 we set the filtering problem with signal dependent observation noise and the framework for transforming this singular filtering problem into the classi-cal one. Then, we discuss this transformation in two cases. In Section 3 we consider the case when the signal dimension is small so (under mild regularity) level set Mz = {x ∈Rd : σ(x) = z} is discrete for each positive definite matrixz. In Section 4 we study the transformation when Mz are manifolds. In this case, we decompose the vectors inRd into their tangent and normal components, and study the signal according to this decomposition. In Section 5, we convert the filtering with OU-noise to a special case of the general singular filtering model.
The methods and results presented here benefited a lot from the work of Ioannides and LeGland (see
[6]and[7]). In particular, in[7], they study the filtering problem with perfect observations. That is, in their set-up, the observation processY is adeterministicfunction of the signal X. Here we show that the filtering problem that we are interested in can be reduced to one where the observation process has two components: one that is perfect (in the language of Ioannides and LeGland ) and one that is of the classical form (see Lemma 2.3 below).
2
A general singular filtering model
Motivated by the filtering problem with OU-process as noise, we will consider a general filtering problem with signal and observation given by
¨
d Xt=b(Xt)d t+c(Xt)dWt+˜c(Xt)d Bt
where B andW are two independent Brownian motions in Rd andRm respectively, b, c, ˜c, h, σ
are functions defined onRd with values inRd,Rd×m,Rd×d,Rm,Rm×m, respectively. We will assume thatY0=0 and thatX0is independent ofBandW. We will denote the law ofX0 byπ0 and assume
that it is absolutely continuous with respect to the Lebesgue measure onRd and will denote by ˜π0
its density.
We will also assume thatσ(x)is a symmetric for each x and positive definite matrix for each x. If not, we can define them-dimensional process ˜W
dW˜t =pσσT(X t)
−1
σ(Xt)dWt, t≥0.
Then the pair process(W˜,B)is a standardm+d-dimensional Brownian motion and the system (2.1) is equivalent to the following
¨
d Xt=b(Xt)d t+c(Xt)dW˜t+˜c(Xt)d Bt d Yt=h(Xt)d t+pσσT(X
t)dW˜t.
The analysis can be easily extended to cover the case when all terms in (2.1) depend on bothX and Y. We make the following assumptions throughout the rest of the paper.
Condition (BC): The functions b, c, ˜c are Lipschitz continuous and h is bounded and measurable. Condition (ND): For any x∈R, the m×m-matrixσ(x)in invertible.
Condition (S): The partial derivatives of the functionσup to order 2 are continuous.
Condition (X0): The law of X0has a continuous densityπ˜0with respect to the Lebesgue measure onRd.
Let〈Y〉be the quadratic covariation process of them-dimensional semimartingaleY. From (2.1) it follows that
〈Y〉t =
Z t
0
σ2 Xt
ds.
Hence, the processσ2(Xt) = d td 〈Y〉tisFtY-measurable for anyt>0.
Remark 2.1. σ2(X0)may not beF0Y-measurable, but isF0Y+-measurable.
We decompose the observationσ-fieldFY
t into two parts: One is generated byσ(Xt)and another by an observation process of the classical form.
Denote by Sm+ the set of symmetric positive definite m×m-matrices. Then for x ∈Rd, we have
σ(x) and σ2(x) ∈ S+
m. Next, we define the mapping a from S +
m to Rm˜ with ˜m =
m(m+1)
2 as
the list of the diagonal entries and those above the diagonal in lexicographical order, i.e., for any r∈ Sm+, a(r)is defined as
a(r)1=r11, a(r)2=r12, ..., a(r)m=r1m, a(r)m+1=r22, ..., a(r)2m−1=r2m,
a(r)2m=r33, ..., a(r)m˜=rmm.
Lemma 2.2. a(Sm+)is open inRm˜.
Proof: Supposeσ∈Rm×m is symmetric. Then,σ∈ Sm+if and only ifd et(σk)>0, k=1, 2,· · ·,m, where σk is the k×k sub-matrix obtained from σ by removing the last m−k rows and m−k columns. Note thatd et(σk)is a polynomial of the entries inσ. Thus the image a(Sm+)consists of points inRm˜ such that these polynomials of its coordinates are positive. This implies thata(Sm+)is open.
Now letsbe the square root mappings:Sm+ → Sm+ such that for anyr ∈ Sm+, s(r)is the unique matrix belonging toSm+such thats(r)2=r. It is easy to see thatsis a continuous and, in particular, a Borel measurable mapping. Hence, sinceσ2(Xt)isFtY-measurable for anyt>0, then
Zt =σ(Xt) =s(σ2(Xt)), t>0
isFY
t -measurable for anyt>0, too. We have the following Lemma 2.3. LetY be the stochastic process defined byˆ
dYˆt=σ−1(Xt)d Yt=˜h(Xt)d t+dWt,
where˜h=σ−1h. Then,
FtY =FtYˆ∨ FtZ, t>0.
Proof: From the aboveFZ
t ⊂ FtY. Alsoσ
−1(X
t)isFtZ-measurable, henceYˆtisFtY-measurable, i.e.,
FtYˆ⊂ FtY. So
FtYˆ∨ FtZ ⊂ FtY.
On the other hand, asd Yt =ZtdYˆt, we have indeed,
FtY ⊂ FtYˆ∨ FtZ.
By Lemma 2.3,Z can be considered as part of the observations and the signal-observation pair can be written as
d Xt=b(Xt)d t+c(Xt)dWt+˜c(Xt)d Bt d Yt=h(Xt)d t+ZtdWt
Zt=σ(Xt)
. (2.2)
We see now that the framework is truly non-classical as part of the observation process is noiseless. It follows that, given the observation,Xttakes values in the level setMZt as defined in the introduction. Hence,πt has support onMZ
t. Therefore,πt will not have total support (unlessσis constant) and
will be singular with respect to the Lebesgue measure onRd.
As was seen, only the diagonal entries and those above the diagonal of the process Zt (in other words,a Zt
) are required to generateFtZ. Hence, we only need to take into account the properties of the mappingaσ
defined as aσ(x) = a(σ(x)) for all x ∈Rd. Then, defining the usual gradient operator for Rm˜ -vectors f by
∇f
(x)i j=∂jfi(x); i=1, ..., ˜m; j=1, ...,d; x ∈Rd;
q(x) =∇aσ(x) is a linear mapping fromRd toRm˜, i.e., an ˜m×dmatrix.
Definition 2.4. The vector x∈Rdis a regular point for aσif and only if the matrix q(x)has full rank min(d, ˜m). We shall denote the collection of all regular points byR.
We will study the optimal filterπt in the next two sections according to the type of the level setMz.
3
The case
d
≤
m
˜
In this section, we consider the case when d ≤ m˜. We will assume that Mz consists of countably many points and that its connected components do not branch or coalesce. Namely, we assume
Condition (R1): There exists a countable set I such that for any z∈Rm˜,
Mz={xi(z): i∈I},
where xi:Rm˜ →Rd, i∈I are continuous functions such that xi(z)6= xj(z), for any i,j∈I , i6= j.
This condition holds true ifaσsatisfies the following assumption
Condition (R): Every point inRd is regular for aσ, in other words,R=Rd.
In this case, for x ∈ Rd, by the inverse function theorem (see, for example, Rudin [12]) there is a continuous bijection between an open neighborhood of w(aσ(x))and an open neighborhood of x, where w : Rm˜ → Rd is the projection of Rm˜ onto Rd corresponding to those coordinates which give a minor of maximal rank for the matrixq(x). The composition between this continuous bijection and the projectionwis, in effect, one of the continuous functionsxi appearing in Condition (R1). In particular, if x ∈Mz, then there is no other element of Mz within the corresponding open neighborhood. In other words,Mzcontains only isolated points. Thus, Condition (R) implies that all the level sets ofσ(respectively,aσ) arediscreteand must be finite on any compact set and therefore countable overall. Hence there is a countable set of continuous functions describing the level sets. These continuous functions do not coalesce or branch as that would contradict the existence of the bijection at the point of coalescence/branching (in topological language, the number of connected components of the level sets is locally constant). Hence condition (R) implies Condition (R1).
By Condition (R1) and the continuity of the process Xt, we see that if X0 is non-random and X0 = xi(Z0) for some i ∈ I, then Xt = xi(Zt) for the same value of i ∈ I (the process does not ‘jump’ between connected components. In other words, the noiseless component of the observation uniquely identifies the conditional distribution of the signal given the observation:
πt=δX
Next, we consider the case that X0 is not constant. We need to take into account the additional information that may arise from observing the quadratic variation of the process a Zt
and the covariation process betweena(Zt)andYt. This will not influence the trajectory ofπ: From above we already know that it is deterministic given its initial valueπ0 and the processa Zt
. However, this additional information may restrictπ0.
Applying Itô’s formula toaσ(Xt)forXt being given by (2.1), we get
where Lis the second order differential operator given by
L f = 1
It follows from (3.1) and (2.1) that the quadratic covariation process betweena(Zt)andYt is
Z t
The analysis that follows hinges upon an explicit characterization of the information contained in
F0Y+. Such a characterization may not be available in general. However, we present two cases where it is possible and note that other cases may be deduced in an inductive manner. From Remark 2.1 we know that σ2(X
0), which is the derivative of the quadratic variation of the process Y at 0, is F0Y+-measurable. Moreover, from the discussion following Lemma 2.2 σ(X0) and aσ(X0) are also F0Y+-measurable. In Case 1, no more additional information is available from ˜Z0. This means that the FY
0+-measurable random variables obtained by differentiating at zero the quadratic variation ofaσ(X)and the quadratic covariation process between aσ(X) andY are functions of aσ(X0). In
Case 2, these two new processes offer new information (they are not functions ofaσ(X0)), but their corresponding quadratic variations and quadratic covariation processes do not have informative derivatives. In subsequent cases, which can be treated in a similar manner as the first two cases, more and more of the processes constructed by computing quadratic variations and quadratic covari-ations and differentiating at zero offer information (they are not functions of the already computed
F0Y+-measurable random variables).
Case 1: The matrices qccT+˜c˜cTqT and qc are functions of aσ, in other words there exist two Borel measurable functionsH1 andH2froma(Sm+)toRm˜×m˜ andRm˜×m respectively such that
qccT+˜c˜cTqT =H1 aσ
Let us note that x∈Rdis a regular point if and only if
brings no new knowledge, hence we can ignore it. Following the proof of Theorem 2.8 in[6], we let µz be the conditional probability distribution ofX0 givenZ0=z, i.e.,
where Hd is the d-dimensional Haussdorff measure (cf. Evans and Gariepy [2], p60). Taking
Thus
Following the case with constantX0, we then have
πt f
Before we consider the second case, we give an example for which the conditions of Theorem 3.1 are satisfied.
whereZdenotes the collection of integers and the continuous functions xi:R3→Rare defined as
xi(z) =xi(z1,z2,z3) =
¨
2πi+arcsin(z1−2) z3≥2
2πi+π+arcsin(2−z1) z3<2
.
This proves Condition (R1). Condition (3.3) is also satisfied as
q(x) =
Now we consider the other case.
Case 2: IfqccT +˜c˜cTqT and qc are not functions of aσ, then ˜Z0 as defined in (3.2) offers new information, that is F0Y+ is larger than the σ-field generated by Z0. Hence the support of the distribution ofX0 givenF0Y+may be smaller than givenσ(Z0). To handle the new information, we
need to impose additional constraints. Letσ0 =σ,q0 =qandσk be the following matrix valued function
σk(x) =
σk−1(x) cTqkT−1(x)
qk−1c(x) qk−1
ccT+˜c˜cTqkT−1(x)
, x ∈Rd, k=1, 2, 3..., (3.5)
˜
mk be the dimension of the image of the mappingaσk andqk=∇aσk. We replace Conditions (S),
(R1) and (3.3) with the following:
Condition (˜S): The partial derivatives ofσ1up to order 2 are continuous.
Condition (˜R1): There exists a countable set˜I such that for any z∈Rm˜1,
Mz≡ {x ∈Rd : σ1(x) =z}={xi(z): i∈˜I},
where xi:Rm˜1→Rd, i∈˜I are continuous functions such that xi(z)6=xj(z), for any i,j∈˜I , i6= j.
Condition (INk): qkc and qk
ccT+˜c˜cTqkT are functions of aσk.
Then, the following analogue of Theorem 3.1 holds true.
Theorem 3.3. Suppose that the conditions (˜R1, ˜S, BC, X0, IN1) are satisfied. If Px∈M˜
Z0
˜
π0(x) J(x) <∞, then
πt= X
x∈M˜Z 0
pxδXˆx
t, t>0,
whereXˆtx =xi(Zt)with i∈˜I is such that x= xi(Z0), and
px = ˜
π0(x) J(x)
P
x∈M˜Z 0
˜
π0(x) J(x)
.
We now give an example for which the conditions of Theorem 3.3 are satisfied.
Example 3.4. Let d = m =1. The coefficients of the system are b(x) = x, c(x) = 1, ˜c = 1, h(x)
bounded, andσ(x) =2+sinx. Then, q(x) =cosx is not a function ofσ. However, q′is a function of
σas
q′(x) =−sinx =2−σ(x).
In this casem˜1=3,
σ1(x) =
2+sinx cosx cosx 2(cosx)2
q1(x) =
cosx
−sinx 4 cosxsinx
, x ∈R
is a function of aσ
1 so (IN1) holds and the level sets Mz are described by
Mz={xi(z): i∈Z},
whereZdenotes the collection of integers and the continuous functions xi:R3→Rare defined as
xi(z) =xi(z1,z2,z3) =
¨
2πi+arcsin(z1−2) z2≥0 2πi+π+arcsin(2−z1) z2<0 .
This proves Condition (˜R1). The other conditions are easy to verify.
The analysis can continue in this manner: if (IN1) is not satisfied by q1, we can define σ2 in a
similar manner withσ,q in (3.5) replaced by σ1 andq1 respectively; and the above procedure is then continued until Condition (INk) is satisfied.
4
The case
d
>
m
˜
In this section, we consider the case whend>m˜. We will show that Mz is no longer a discrete set but rather a surface (manifold) and the optimal filterπt is a probability measure on the manifold Mz and is absolutely continuous with respect to the surface measure. For this, we follow closely the analysis in[7].
Note that, ifd>m˜, then x∈Rd is a regular point if and only if
J(x) =pdetqqT(x)>0.
We shall useTxMzto denote the space consisting of all the tangent vectors to the manifold (surface) Mz at point x ∈Mz. TxMz is called the tangent space of Mz at x. We will useNxMz to denote the orthogonal complement of TxMz in Rd. We call NxMz the normal space of Mz at x. We list in the following lemma some well-known facts about the transformationaσwithout giving their proofs:
Lemma 4.1. i) For any u∈Rm˜, Ma−1(u)is ad-dimensional manifold where˜ d˜=d−m. The Haussdorff˜ measureHd˜on Ma−1(u)is the surface measure.
ii) For any x ∈Mz, the rows of the matrix q generate the normal space NxMz to the manifold Mz at point x.
iii) Let
ρ(x) =qT(qqT)−1(x)and p(x) =ρ(x)q(x).
Then p(x)is the orthogonal projection matrix fromRm˜ to the subspace NxMz.
For simplicity of the presentation, we make an assumption which is slightly stronger than (3.3).
Condition (IN): There exist two Borel measurable functions H1and H2from a(S+m)toRm˜×mandRm˜×d
respectively such that
qc=H1 aσ
andq˜c=H2(aσ). (4.1)
Example 4.2. Let d = 2and m = 1. Define the coefficients by b(x) = x, c(x) =0, ˜c(x) = I and
σ(x) =ex1. Then J(x) =ex1>0for all x∈R2and hence, Condition (R) is satisfied. Further,
Mz={(x1,x2): ex1=z}
is a (line) manifold of dimensiond˜=2−1=1. It is clear that the row vector of q generates the normal space
NxMz={λ(1, 0): λ∈R}.
In this example, p(x) =
1 0 0 0
is the orthogonal projection matrix fromR2 to NxMz. To verify
(IN), we note that qc=0and
q˜c= (ex1, 0)I= (σ(x), 0).
We also assume:
Condition (C): Both c and˜c are continuously differentiable.
Throughout the rest of this section, the assumptions (R, S, BC, X0, IN, ND, C) will be in force. The following theorem gives us the conditional distribution of X0 given Z0. Since, in this case, F0Y+
coincides with theσ-field generated by Z0, it gives, in effect,π0+in the case ofd>m˜.
We introduce λu to be the surface measure on the level set Ma−1(u) foru ∈Rm˜ and µz to be the conditional probability distribution ofX0 givenZ0=z, i.e.,
µz(d x) =P X0∈d x|Z0=z
.
The following theorem shows thatµz is absolutely continuous with respect toλu.
Theorem 4.3. Suppose that the densityπ˜0is not identically zero on Ma−1(u)and satisfies the following integrability condition:
Z
Ma−1(u) ˜
π0(x)
J(x) λu(d x)<∞.
Then
µz(d x) =p(x)λa(z)(d x),
where
p(x) = R π˜0(x)/J(x)
Mzπ˜0(y)/J(y)λu(d y)
.
Proof: For any test functionφdefined onRd, and any Borel setDinRm×m, define
Then, by the co-area formula (cf. Evans and Gariepy[2], chapter 3), we have
The result follows from the definition of the conditional expectation.
We now decompose the vector fields in the SDE satisfied by the signal according to their components in the spacesTxMz andNxMz. It is more convenient to use Stratonovich form for the signal process. That is, the signalX satisfies the following SDE in Stratonovich form:
d Xt=˜b(Xt)d t+c(Xt)◦dWt+˜c(Xt)◦d Bt, (4.2)
Recall from (3.1) that theFtY-measurable processa(Zt)satisfies
d a(Zt) =Laσ(Xt)d t+d Vt, (4.3)
whereVt is the ˜m-dimensional continuous martingale
Lemma 4.4. The second observation process of (2.2) a(Zt)satisfies
d a(Zt) =qc(Xt)◦dWt+q˜c(Xt)◦d Bt+q˜b(Xt)d t. (4.4)
Proof: We have
d Vt = qc(Xt)◦dWt+q˜c(Xt)◦d Bt−1
2 d
X
k,j=1
m
X
ℓ=1
∂k(q·jcjℓ)ckℓ(Xt)d t
−1
2 d
X
k,j,ℓ=1
∂k(q·j˜cjℓ)˜ckℓ(Xt)d t
= qc(Xt)◦dWt+q˜c(Xt)◦d Bt− 1 2
d
X
k,j=1
(ccT)jk∂k j2aσd t− 1 2
d
X
k=1
q∂kcc·Tkd t
−1
2 d
X
k,j=1
(˜c˜cT)jk∂k j2aσd t− 1 2
d
X
k=1
q∂k˜c˜c·Tkd t
= qc(Xt)◦dWt+q˜c(Xt)◦d Bt−Laσ(Xt)d t+q˜b(Xt)d t.
Combining with (4.3), we see that (4.4) holds.
Finally, we arrive at the main decomposition result.
Theorem 4.5. Under Condition (BC, S, R, X0, C), the filtering model (2.1) is rewritten as
d Xt= (I−p)˜b(Xt)d t+c(Xt)◦dWt+˜c(Xt)◦d Bt+ρ(Xt)◦d a(Zt), (4.5)
with observations
d a(Zt) =Laσ(Xt)d t+d Vt, (4.6)
and
d Yt=h(Xt)d t+ZtdWt. (4.7)
Proof: From equation (4.2) we have
d Xt = ˜b(Xt)d t+c(Xt)◦dWt+˜c(Xt)◦d Bt
= (I−p)˜b(Xt)d t+ (I−p)c(Xt)◦dWt+ (I−p)˜c(Xt)◦d Bt
+p˜b(Xt)d t+pc(Xt)◦dWt+p˜c(Xt)◦d Bt. (4.8)
By (4.4), we get
ρ(Xt)◦d a(Zt) =p˜b(Xt)d t+pc(Xt)◦dWt+p˜c(Xt)◦d Bt.
Plugging back into (4.8), we see that (4.5) holds. The equality (4.6) is a rewriting of (4.3). The equation (4.7) is just the original observation model (2.1).
Let{ξt,s: 0≤s≤t}be the stochastic flow associated with the SDE:
Lemma 4.6. The flowξt,s maps MZ
s to MZt. Further, the processξt isF
Z
t -adapted.
Proof: Applying the Stratonovich form of Itô’s formula, we get
d aσ(ξt) =∇Taσ(ξt)ρ(ξt)◦d a(Zt) =d a(Zt).
Thus forσ(ξs) =Zs, we getσ(ξt) = Zt. The second conclusion follows from the uniqueness of the solution to the SDE (4.9).
Denote the column vectors of(I−p)c and(I−p)˜cby g1, · · ·, gmand ˜g1, · · ·, ˜gd respectively. Let b0= (I−p)˜b. Then for each x ∈Mz, the vectors g1(x), · · ·, gm(x); ˜g1(x), · · ·, ˜gd(x); b0(x)are
all inTxMz. The signal processXt satisfies
d Xt =ρ(Xt)◦d a(Zt) +
m
X
i=1
gi(Xt)◦dWti+
d
X
j=1
˜
gj(Xt)◦d Btj+b0(Xt)d t. (4.10)
It is well-known that the Jacobian matrixξ′t,s of the stochastic flowξt,s is invertible (cf. Ikeda and Watanabe[5]). The operator(ξ−t,1s)∗defined below pulls a vector atξt,s(x)back to a vector atx.
Definition 4.7. Let g be a vector field inRd. The random vector field(ξ−t,s1)∗g is defined as
(ξ−t,1s)∗g(x)≡(ξ′t,s)−1g(ξt,s(x))
for any aσregular point x∈Rd.
We consider the following SDE onRd:
dκt= (ξ−t,01)∗b0(κt)d t+ m
X
i=1
(ξ−t,01)∗gi(κt)◦dWti+
d
X
j=1
(ξ−t,01)∗˜gj(κt)◦d Btj. (4.11)
Lemma 4.8. The SDE (4.11) has a unique strong solution. Further, if Z0 = z, thenκt ∈Mz for all t≥0a.s.
Proof: Note that(ξ′t,0)−1satisfies a diffusion SDE with bounded coefficients (cf. Ikeda and Watanabe
[5]). By Gronwall’s inequality, it is easy to show that
E sup
0≤t≤T
k(ξ′t,0)−1kp<∞, ∀p>1.
Therefore, we can prove that there is a constantKsuch that
E (ξ
−1
t,0)∗b0(κ1)−(ξt−,01)∗b0(κ2)
2
≤K|κ1−κ2|2, ∀κ1, κ2∈Rd.
Applying Itô’s formula to (4.11), we get
d aσ(κt) = q(κt)(ξ−t,01)∗b0(κt)d t+ m
X
i=1
q(κt)(ξ−t,01)∗gi(κt)◦dWti
+
d
X
j=1
q(κt)(ξ−t,01)∗˜gj(κt)◦d Btj.
Sinceb0(ξt,0(κt))∈Tξt,0(κt)MZt, we have(ξ
−1
t,0)∗b0(κt)∈TκtMσ(κt). As the row vectors ofq(κt)are
inNκ
tMσ(κt), we see that
q(κt)(ξ−t,01)∗b0(κt) =0.
Also the above equality holds withb0replaced bygi, 1≤i≤mor ˜gj, 1≤ j≤d. Thus,d aσ(κt) =0, and hence,aσ(κt) =aσ(κ0),∀t≥0. This proves thatσ(κt) =z, and henceκt ∈Mz,∀ t≥0, a.s.
The next theorem gives the decomposition of the signal process.
Theorem 4.9. For almost allω∈Ω, we have
Xt(ω) =ξt,0(κt(ω),ω), ∀t≥0. (4.12)
Proof: Denote the right hand side of (4.12) by ˜Xt(ω). Applying Itô’s formula, we get
dX˜t =
d
X
j=1
∂jξt,0(κt)◦dκjt+ρt(ξt,0(κt))◦d a(Zt)
= ξ′t,0(κt)(ξ′t,0(κt))−1b0(ξt,0(κt))d t+
m
X
i=1
ξ′t,0(κt)(ξ′t,0(κt))−1gi(ξt,0(κt))◦dWti
+
d
X
j=1
ξ′t,0(κt)(ξ′t,0(κt))−1˜gj(ξt,0(κt))◦d B j
t+ρt(X˜t)◦d a(Zt)
= ρ(X˜t)◦d a(Zt) + m
X
i=1
gi(X˜t)◦dWti+ d
X
j=1
˜
gj(X˜t)◦d B j
t+b0(X˜t)d t.
By the uniqueness of the solution to (4.10), we see that the representation (4.12) holds.
The optimal filter then satisfies
πtf =E(f(ξt,0(κt))|FtYˆ∨ FtZ), t>0.
Note that ξt,0 is FtZ-measurable. Thus, we may regard ξt,0 as known and the singular filtering problem can be transformed to a classical one as follows: For f ∈Cb(Mz), let
Utf ≡E[f(κt)|F ˆ Y t ∨ F
Z t ].
ThenUt is the optimal filter with the signal processκt being given by (4.11) and the observation
( ˆYt,a(Zt))being given by
and
d a(Zt) =Laσ(ξt,0(κt))d t+H1(a(Zt))dWt+H2(a(Zt))d Bt. (4.14)
Note that the filtering problem with signal (4.11) and observations (4.13) and (4.14) is classical. We sketch below the derivation of the Zakai equation for the unnormalized version ˜Ut of the process Ut, and leave the details to the reader. Once this is done, we note that
πtf =Ut
Then, ˜Zt is observable and
dZ˜t=˜h(t,Y,κt)d t+d
Note that (4.11) can be written in the Itô form
for suitably defined coefficients ˜b, ˜σ1 and ˜σ2. Denote
ˆ
σ(t,Y,κt) =
˜
σ1(t,Y,κt), ˜σ2(t,Y,κt)h2(a(Zt))
H2(a(Zt))H2(a(Zt))T
1
2
and
Lt,Yf(κ) = 1 2
X
i j ˜
ai j(t,Y,κ)∂i jf(κ) +˜b(t,Y,κ)T∇f(κ), (4.17)
where ˜a(t,Y,κ) =P2i=1σ˜i(t,Y,κ)σ˜i(t,Y,κ)T. Then, by Kallianpur-Striebel formula, we have
Ut=U˜t1−1U˜t, (4.18)
while the unnormalized filter ˜Ut satisfies the following Zakai equation:
dU˜tf =U˜t(Lt,Yf)d t+U˜t
∇Tfσ(ˆ t,Y) +˜h(t,Y)TfdZ˜t. (4.19)
A filtering equation forUt can be derived by applying Itô’s formula to (4.18) and (4.19). Now we summarize the discussion above to a theorem.
Theorem 4.10. Suppose that the assumptions (R, S, BC, X0, IN, ND) are satisfied. Let Z be the˜ observation process given by (4.15). Let˜h and Lt,Y be given by (4.16) and (4.17) respectively. Let
ξt,s be the stochastic flow given by (4.9). Let U˜t be the uniqueMF(MZ0)-valued solution to the Zakai equation (4.19) and Ut = (U˜t1)−1U˜
t. Then, the optimal filterπtis given by
πt(f) =Ut(f ◦ξt,0), ∀ f ∈Cb(Rd).
Finally, we indicate that an analogue of the Kalman filter for the singular filtering setup is a part of a work in progress by Liu[10]. We indicate the model and the result anticipated. For simplicity, we takem=1.
Let the signal be given by the linear equation:
d Xt= bXtd t+d Bt,
and the observation is
d Yt =hTXtd t+|QTXt|dWt,
where (B,W)is the d+1-dimensional Brownian motion, b is ad×d-matrix andh, Q∈Rd. It is anticipated thatπt will be a linear combination of two normal measures on planes{x ∈Rd: QTx = ±Zt}, whereZt=|QTXt|is observable. Equations for the condition means and variances, as well as the weight process in the linear combination, will be derived.
5
The filtering model with Ornstein-Uhlenbeck noise
to a singular filtering problem of the form studied in the previous sections. To be more general, we assume that the OU process is given by
dOt=a1Otd t+b1dWt
and y is as in (1.4). Let
Yt =
Z t
0
e−a1sd(ea1sy s).
ThenFY t =F
y
t and, by Itô’s formula,
d Yt= (Lh(Xt) +a1h(Xt))d t+b1dWt+∇hc(Xt)d Bt.
Define the pair process
d Vt = (b21I+∇hc(∇hc)T)−1/2(Xt) b1dWt+∇hc(Xt)d Bt
,
dV˜t = (b21I+ (∇hc)T∇hc)−1/2(Xt)(∇hc)T(Xt)dWt−b1d Bt.
Then the pair process(V, ˜V)is an (m+d)-dimensional Brownian motion and
d Bt = (b21I+∇hc(∇hc)
T)−1(∇hc)T(b2
1I+∇hc(∇hc)
T)−1/2(X
t)d Vt
−(b12I+ (∇hc)T∇hc)−1/2(Xt)dV˜t.
Hence, if we define the following functions:
σ = (b21I+∇hc(∇hc)T)1/2
c1 = c(b21I+ (∇hc)
T∇hc)−1(∇hc)T(b2
1I+∇hc(∇hc)
T)1/2
˜c1 = −c(b12I+ (∇hc)T∇hc)−1/2,
then the pair signal/observation process is written as
¨
d Xt=b(Xt)d t+c1(Xt)d Vt+˜c1 Xt
dV˜t d Yt= Lh+a1h
(Xt)d t+σ(Xt)d Vt . (5.1)
Hence the filtering model (5.1) is of the form (2.1).
Next, we give an example to demonstrate that for filtering problem with OU observation noise, both discrete and continuous singularity can occur. This example is a special case of those considered by Liu[10], so we omit the details here.
Example 5.1. Let the signal Xit, i=1, 2, be governed by SDEs
d Xti =σiÆXtid Bit,
where σ1 and σ2 are two constants, and B1t and B2t are two independent one-dimensional Brownian motions. Suppose that the observation model is
yt=X1t +X
2
t +Ot,
Applying the quadratic variation procedure indicated above, we see that Zt ≡ σ21Xt1+σ22Xt2 is observable. It can be proved that when σ1 = σ2, the singular filter is of continuous type. In this case, the additional condition (IN) in Section 4 is satisfied after transforming the filtering problem with OU noise to a singular filtering problem.
Whenσ16=σ2, the condition (IN) is not satisfied. In this case,
Ztp≡σ21pX1t +σ22pX2t, p=1, 2,· · ·
are all observable and hence the level sets are countable and the conditional distribution is of dis-crete type.
Finally, we would like to point out that the linear filtering with OU process as the observation noise has been studied by Liu[10]. The model is given by
¨
d Xt= (b1Xt+b0)d t+C d Bt, yt=H Xt+Ot,
whereOt is them-dimensional OU-process independent ofBt, H, b1, b0, C are matrices of dimen-sions m×d, d×d, d×1,d×d, respectively. It is classified to three classes (classical, continuous singular, discrete singular) according to the ranks of some matrices determined by the coefficients H, b1, b0, C. We refer the interested reader to the working paper[10]for detail.
Acknowledgement. The authors would like to thank the referee for helpful suggestions.
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